CoParenting: kids belongings

Question: How do coparents deal with shared custody on clothing? I have always sent my son to his Dad with clothing, toys, hygiene, gadgets etc but whenever his Dad (once in a blue moon) buys him ANYTHING, he tells my son he is not allowed to take it with him to mom’s. It bothers me, like why? I have never cared of what article of clothing my son takes or brings nor toys or gadgets. Does that bother anyone else? If I were to tell my son, “you cannot take anything I have bought you to your Dads” he would take nothing.